Question: All of the 5th grade teachers and students from Gardner Bullis went on a field trip to an art museum. Tickets were $$8.50$ each for teachers and $$3.00$ each for students, and the group paid $$64.00$ in total. The next month, the same group visited a science museum where the tickets cost $$34.00$ each for teachers and $$12.50$ each for students, and the group paid $$261.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+3y = 64}$ ${34x+12.5y = 261}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-34x-12y = -256}$ ${34x+12.5y = 261}$ Add the top and bottom equations together. $ 0.5y = 5 $ $ y = \dfrac{5}{0.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {8.5x+3y = 64}$ to find $x$ ${8.5x + 3}{(10)}{= 64}$ $8.5x+30 = 64$ $8.5x = 34$ $x = \dfrac{34}{8.5}$ ${x = 4}$ You can also plug ${y = 10}$ into $ {34x+12.5y = 261}$ and get the same answer for $x$ ${34x + 12.5}{(10)}{= 261}$ ${x = 4}$ There were $4$ teachers and $10$ students on the field trips.